III. Optics law for a sound-wave in a medium where the speed is variable.

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The sound-waves change of direction of propagation when they change of medium towards a one with a different speed .

The law to apply to calculate the new direction is the well-known formula:

where C is the velocity and n=Co/C the index of refraction with

Co= 299,792,458 m/s. This formula is easily demonstrated by application of the Huygens's principle.

Now we are going to calculate the change of direction when several layers of different speeds are crossed (figure below).

In the medium of index N1, the vectorial speed of the sound-wave is:

In the medium of index N2 we have:

In the medium of index N3, we have:

And thus the last formula is the same in the N4, N5,...media in replacing N3 by N4, N5,etc...

It is clear here, that if the number of layers between the medium with the index N1 and the medium of index N(x) increases to infinity, the equation obtained will not change at all.

That means that the equation is valid for a continuously variable index in the medium.

Now we consider only the X component of the (sound-wave) light velocity, that is to say:

, we have:

To obtain that formula, the reader must note that: and

.

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